Question

Compute the following using the residue theorem (complex analysis):

$\int_{0}^{2\pi} \frac{1}{1+cos^{2}\theta} d\theta$

Answer 1

2TT I do 1 + cos²o using residue Theorem lo e Coso = + e 2 let Z=elo z tz 2 dz= i elo do 10,0) unit circle with control dz= iz do نه ما dz iz 1+12+ z 12 +2 472 dz i z 2 1 + (22+1 22 je S. dz uz 422 +241 +22² dz uz (24 + 6 2² +1) 2: Now Let u=z2 0 2T ОУЧ7г (twice asound () du= 2zdz DOO

- Od 2 du ilu²+6u+1) becouse 2 times unit circle. De u du 1 (u²+ butl) Now poles u2+ 64+1 u= -6 + 536-4 2 u= - 6 t 32 2 u = -3 + 252 Uz -3-252 inside circle outside ciode Now using Residul Theorem Residue at u= -3 +252 (-3+ 252) im u- -3 +252 u2+ out V um U7 -3 +252 24+6 Residus yuz - 6+452 +6 du = 25i ( Residue) = 21 (u²+ butt) 2ti s 452 sola (27) los (4246utl) i 2 T Ars vo